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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Thin stationary sets and disjoint club sequences

Authors: Sy-David Friedman and John Krueger
Journal: Trans. Amer. Math. Soc. 359 (2007), 2407-2420
MSC (2000): Primary 03E35, 03E40
Published electronically: December 5, 2006
MathSciNet review: 2276627
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Abstract: We describe two opposing combinatorial properties related to adding clubs to $ \omega_2$: the existence of a thin stationary subset of $ P_{\omega_1}(\omega_2)$ and the existence of a disjoint club sequence on $ \omega_2$. A special Aronszajn tree on $ \omega_2$ implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of $ \omega_2$ which cannot acquire a club subset by any forcing poset which preserves $ \omega_1$ and $ \omega_2$. We prove that the existence of a disjoint club sequence follows from Martin's Maximum and is equiconsistent with a Mahlo cardinal.

References [Enhancements On Off] (What's this?)

  • 1. U. Abraham and S. Shelah, Forcing closed unbounded sets, Journal of Symbolic Logic 48 (1983), no. 3, 643-657. MR 0716625 (85i:03112)
  • 2. J. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions I, Transactions of the American Mathematical Society 270 (1982), no. 2, 557-574. MR 0645330 (83k:03040a)
  • 3. Q. Feng, T. Jech, and J. Zapletal, On the structure of stationary sets, Preprint.
  • 4. M. Foreman, M. Magidor, and S. Shelah, Martin's maximum, saturated ideals, and non-regular ultrafilters. part I, Annals of Mathematics 127 (1988), 1-47. MR 0924672 (89f:03043)
  • 5. S. Friedman, Forcing with finite conditions, Preprint.
  • 6. M. Gitik, Nonsplitting subset of $ P_\kappa \kappa^+$, J. Symbolic Logic 50 (1985), no. 4, 881-894. MR 0820120 (87g:03054)
  • 7. W. Mitchell, $ I[\omega_2]$ can be the non-stationary ideal, Preprint.
  • 8. William Mitchell, Aronszajn trees and the independence of the transfer property, Ann. Math. Logic 5 (1972/73), 21–46. MR 0313057,
  • 9. M. Rubin and S. Shelah, Combinatorial problems on trees: Partitions, $ \Delta$-systems and large free subtrees, Annals of Pure and Applied Logic 33 (1987), 43-81. MR 0870686 (88h:04005)
  • 10. Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206

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Additional Information

Sy-David Friedman
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Waehringer Strasse 25, A-1090 Wien, Austria

John Krueger
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

Received by editor(s): June 28, 2005
Published electronically: December 5, 2006
Additional Notes: The authors were supported by FWF project number P16790-N04.
Article copyright: © Copyright 2006 American Mathematical Society

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